Mathematics Commons^™

Totally Isotropic Subspaces Of Small Height In Quadratic Spaces, Wai Kiu Chan, Lenny Fukshansky, Glenn Henshaw

CMC Faculty Publications and Research

Let K be a global field or Q, F a nonzero quadratic form on KN , N ≥ 2, and V a subspace of KN . We prove the existence of an infinite collection of finite families of small-height maximal totally isotropic subspaces of (V, F) such that each such family spans V as a K-vector space. This result generalizes and extends a well known theorem of J. Vaaler [16] and further contributes to the effective study of quadratic forms via height in the general spirit of Cassels’ theorem on small zeros of quadratic forms. All bounds on height are …

On An Effective Variation Of Kronecker's Approximation Theorem, Lenny Fukshansky

CMC Faculty Publications and Research

Let Λ ⊂ Rn be an algebraic lattice, coming from a projective module over the ring of integers of a number field K. Let Z ⊂ Rn be the zero locus of a finite collection of polynomials such that Λ |⊂ Z or a finite union of proper full-rank sublattices of Λ. Let K1 be the number field generated over K by coordinates of vectors in Λ, and let L1, . . . , Lt be linear forms in n variables with algebraic coefficients satisfying an appropriate linear independence condition over K1. For each ε > 0 and a ∈ Rn, …

Topological Data Analysis For Systems Of Coupled Oscillators, Alec Dunton

HMC Senior Theses

Coupled oscillators, such as groups of fireflies or clusters of neurons, are found throughout nature and are frequently modeled in the applied mathematics literature. Earlier work by Kuramoto, Strogatz, and others has led to a deep understanding of the emergent behavior of systems of such oscillators using traditional dynamical systems methods. In this project we outline the application of techniques from topological data analysis to understanding the dynamics of systems of coupled oscillators. This includes the examination of partitions, partial synchronization, and attractors. By looking for clustering in a data space consisting of the phase change of oscillators over a …

Line-Of-Sight Pursuit And Evasion Games On Polytopes In R^N, John Phillpot

HMC Senior Theses

We study single-pursuer, line-of-sight Pursuit and Evasion games in polytopes in $\mathbb{R}^n$. We develop winning Pursuer strategies for simple classes of polytopes (monotone prisms) in Rⁿ, using proven algorithms for polygons as inspiration and as subroutines. More generally, we show that any Pursuer-win polytope can be extended to a new Pursuer-win polytope in more dimensions. We also show that some more general classes of polytopes (monotone products) do not admit a deterministic winning Pursuer strategy. Though we provide bounds on which polytopes are Pursuer-win, these bounds are not tight. Closing the gap between those polytopes known to be …

Fibonomial Tilings And Other Up-Down Tilings, Robert Bennett

HMC Senior Theses

The Fibonomial coefficients are a generalization of the binomial coefficients with a rather nice combinatorial interpretation. While the ordinary binomial coefficients count lattice paths in a grid, the Fibonomial coefficients count the number of ways to draw a lattice path in a grid and then Fibonacci-tile the regions above and below the path in a particular way. We may forgo a literal tiling interpretation and, instead of the Fibonacci numbers, use an arbitrary function to count the number of ways to "tile" the regions of the grid delineated by the lattice path. When the function is a combinatorial sequence such …

Adinkras And Arithmetical Graphs, Madeleine Weinstein

HMC Senior Theses

Adinkras and arithmetical graphs have divergent origins. In the spirit of Feynman diagrams, adinkras encode representations of supersymmetry algebras as graphs with additional structures. Arithmetical graphs, on the other hand, arise in algebraic geometry, and give an arithmetical structure to a graph. In this thesis, we will interpret adinkras as arithmetical graphs and see what can be learned.

Our work consists of three main strands. First, we investigate arithmetical structures on the underlying graph of an adinkra in the specific case where the underlying graph is a hypercube. We classify all such arithmetical structures and compute some of the corresponding …

Lattices From Tight Equiangular Frames, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj, Deanna Needell

Pomona Faculty Publications and Research

We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular (k,n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n = k + 1 and that there are infinitely many k such that a lattice emerges for n = 2k …

Reducing Math Anxiety In The Secondary Classroom, Haley Scheldorf

Honors Capstones

Capstone submitted as a graduation requirement for the BSU Honors Program.

A Positivity Criterion For The Wave Equation And Global Existence Of Large Solutions, Marius Beceanu, Avy Soffer

Mathematics and Statistics Faculty Scholarship

In dimensions one to three, the fundamental solution to the free wave equation is positive. Therefore, there exists a simple positivity criterion for solutions. We use this to obtain large global solutions to two well-studied energy-supercritical semilinear wave equations, as well as some new results in the subcritical and critical cases.

Optimal Control Of A Perturbed Sweeping Process With Applications To The Crowd Motion Model, Tan Hoang Cao

Wayne State University Dissertations

The dissertation is devoted to the study and applications of a new class of optimal control problems governed by a perturbed sweeping process of the hysteresis type with control functions acting in both play-and-stop operator and additive perturbations. Such control problems can be reduced to optimization of discontinuous and unbounded dif- ferential inclusions with pointwise state constraints, which are immensely challenging in control theory and prevent employing conventional variation techniques to derive neces- sary optimality conditions. We develop the method of discrete approximations married with appropriate generalized differential tools of modern variational analysis to overcome principal difficulties in passing to …

Principal Component Analysis-Based Anatomical Motion Models For Use In Adaptive Radiation Therapy Of Head And Neck Cancer Patients, Mikhail Aleksandrovich Chetvertkov

Wayne State University Dissertations

Purpose: To develop standard and regularized principal component analysis (PCA) models of anatomical changes from daily cone beam CTs (CBCTs) of head and neck (H&N) patients, assess their potential use in adaptive radiation therapy (ART), and to extract quantitative information for treatment response assessment.

Methods: Planning CT (pCT) images of H&N patients were artificially deformed to create “digital phantom” images, which modeled systematic anatomical changes during Radiation Therapy (RT). Artificial deformations closely mirrored patients’ actual deformations, and were interpolated to generate 35 synthetic CBCTs, representing evolving anatomy over 35 fractions. Deformation vector fields (DVFs) were acquired between pCT and synthetic …

Some New Combinatorial Formulas For Cluster Monomials Of Type A Quivers, Ba Nguyen

Wayne State University Dissertations

Lots of research focuses on the combinatorics behind various bases of cluster

algebras. This thesis studies the natural basis of a type A cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formulas for the cluster monomials in terms of globally compatible collections and broken lines. We give bijective proofs of these formulas by comparing with the well-known combinatorial models of the T-paths and of the perfect matchings in a snake diagram.

Ergodicity Of Stochastic Switching Diffusions And Stochastic Delay Systems, Hongwei Mei

Wayne State University Dissertations

This dissertation contains two main parts. The first part focuses on numerical algorithms for approximating the ergodic means of suitable functions of solutions to stochastic differential equations with Markov regime switching. Our main effort is devoted to obtaining the convergence and rates of convergence of the approximation algorithms. The study is carried out by obtaining laws of large numbers and laws of iterated logarithms for numerical approximation to long-run averages of suitable functions of solutions to switching diffusions.

The second part is devoted to stochastic functional differential equations (SFDEs) with infinite delay. This part consists of two main themes. First, …

A Mechanical Investigation Of Second Order Homogeneous Dynamic Equations On A Time Scale, Jacob E. Fischer

Theses, Dissertations and Capstones

This thesis covers the basic aspects of time scale calculus, a branch of mathematics combining the theories of differential equations and difference equations. Using the properties of time scale calculus we analyze a second order homogeneous dynamic equation with constant coefficients, in particular, y ∆∆ − 1 6 y ∆ + 1 8 y = 0. Following the analysis, this problem will be graphically evaluated using Marshall University’s Differential Analyzer, affectionately named Art. A differential analyzer is a machine that mechanically integrates by way of related rates of rotating rods. The process for making the jump between intervals on a …

Simulations Of The Helmholtz Equation At Any Wave Number For Adaptive Grids Using A Modified Central Finite Difference Scheme, Hafiz Abdul Wajid

Turkish Journal of Mathematics

In this paper, a modified central finite difference scheme for a three-point nonuniform grid is presented for the one-dimensional homogeneous Helmholtz equation using the Bloch wave property. The modified scheme provides highly accurate solutions at the nodes of the nonuniform grid for very small to very large range of wave numbers irrespective of how the grid is adapted throughout the domain. A variety of numerical examples are considered to validate the superiority of the modified scheme for a nonuniform grid over a standard central finite difference scheme.

$H$-Admissible Fourier Integral Operators, Chafika Amel Aitemrar, Abderrahmane Senoussaoui

Turkish Journal of Mathematics

We study in this work a class of $h$-admissible Fourier integral operators. These operators are bounded (respectively compact) in $L^{2}$ if the weight of the amplitude is bounded (respectively tends to 0).

Coefficient Bounds For Subclasses Of M-Fold Symmetric Bi-Univalent Functions, Sevtap Sümer Eker

Turkish Journal of Mathematics

In this study, we introduce and investigate two new subclasses of the bi-univalent functions; both $f(z)$ and $f^{-1}(z)$ are m-fold symmetric analytic functions. Among other results, upper bounds for the coefficients $\left a_{m+1}\right $ and $\left a_{2m+1}\right $ are found in this investigation.

Rate Of Change 2, Ruth Dover

A Simple Introduction to Rates

No abstract provided.

Limits5, Ruth Dover

Limits

Limits and continuity.

Limits2, Ruth Dover

Limits

More on limits, both algebraic and graphical, including one-sided limits.